The minute hand of a clock is 2 cm long. Find the area of the face of the clock described by the minute hand between 7 am and 7 : 15 am.

Answer

To find the area of the face of the clock described by the minute hand between 7 am and 7:15 am, we can follow these steps:

### Step 1: Understand the movement of the minute hand
The minute hand of a clock makes a complete revolution (360 degrees) in 60 minutes. Therefore, in 15 minutes, it will move through a certain angle.

### Step 2: Calculate the angle moved by the minute hand
To find the angle moved by the minute hand in 15 minutes, we can use the following formula:
\[ \text{Angle} = \left(\frac{360 \text{ degrees}}{60 \text{ minutes}}\right) \times 15 \text{ minutes} \]
Calculating this gives:
\[ \text{Angle} = 6 \text{ degrees/minute} \times 15 \text{ minutes} = 90 \text{ degrees} \]

### Step 3: Convert the angle to radians
Since the area of the sector is often calculated using radians, we need to convert degrees to radians. The conversion factor is:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
Thus,
\[ \text{Radians} = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians} \]

### Step 4: Calculate the area of the sector
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{1}{2} r^2 \theta \]
where \( r \) is the radius (length of the minute hand) and \( \theta \) is the angle in radians. Here, the radius \( r = 2 \) cm and \( \theta = \frac{\pi}{2} \):
\[ A = \frac{1}{2} \times (2)^2 \times \frac{\pi}{2} \]
Calculating this gives:
\[ A = \frac{1}{2} \times 4 \times \frac{\pi}{2} = \frac{4\pi}{4} = \pi \text{ cm}^2 \]

### Final Answer
The area of the face of the clock described by the minute hand between 7 am and 7:15 am is \( \pi \) cm².

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